Aim: The size of the climatic niche of a species is a major factor determining its distribution and evolution. In particular, it has been proposed that niche width should be associated with the rate of species diversification. Here, we test whether species niche width affects the speciation and extinction rates of three main clades of vertebrates: amphibians, mammals and birds. Location: Global. Methods: We obtained the time-calibrated phylogenies, IUCN conservation status, species distribution maps and climatic data for 2340 species of amphibians, 4563 species of mammals and 9823 species of birds. We computed the niche width for each species as the mean annual temperature across the species range. We estimated speciation, extinction and transition rates associated with lineages with either narrow (specialist) or wide (generalist) niches using phylogeny-based birth–death models. We also tested if current conservation status was correlated with the niche width of species. Results: We found higher net diversification rates in specialist species than in generalist species. This result was explained by both higher speciation rates (for the three taxonomic groups) and lower extinction rates (for mammals and birds only) in specialist than in generalist species. In contrast, current specialist species tended to be more threatened than generalist species. Main conclusions: Our diversification analysis shows that the width of the climatic niche is strongly associated with diversification rates and may thus be a crucial factor for understanding the emergence of diversity patterns in vertebrates. The striking difference between our diversification results and current conservation status suggests that the current extinction process may be different from extinction rates estimated from the whole history of the group.